From Knot Atlas
Jump to: navigation, search






(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a269 at Knotilus!

Link Presentations

[edit Notes on L11a269's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,5,15,6 X22,20,9,19 X18,7,19,8 X6,17,7,18 X16,22,17,21 X20,16,21,15 X8,13,1,14
Gauss code {1, -2, 3, -4, 5, -8, 7, -11}, {4, -1, 2, -3, 11, -5, 10, -9, 8, -7, 6, -10, 9, -6}
A Braid Representative
A Morse Link Presentation L11a269 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^5+t(1)^3 t(2)^4-4 t(1)^2 t(2)^4+3 t(1) t(2)^4-2 t(1)^3 t(2)^3+8 t(1)^2 t(2)^3-7 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-7 t(1)^2 t(2)^2+8 t(1) t(2)^2-2 t(2)^2+3 t(1)^2 t(2)-4 t(1) t(2)+t(2)+t(1)}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial q^{3/2}-3 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 z^5+3 a^7 z^3+4 a^7 z+2 a^7 z^{-1} -a^5 z^7-4 a^5 z^5-7 a^5 z^3-7 a^5 z-3 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-6 a^3 z^3-3 a^3 z+a^3 z^{-1} +a z^5+3 a z^3+2 a z (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+6 a^9 z^7-11 a^9 z^5+10 a^9 z^3-5 a^9 z+6 a^8 z^8-6 a^8 z^6-a^8 z^4+2 a^8 z^2+4 a^7 z^9-3 a^7 z^5-4 a^7 z^3+6 a^7 z-2 a^7 z^{-1} +a^6 z^{10}+10 a^6 z^8-21 a^6 z^6+17 a^6 z^4-9 a^6 z^2+3 a^6+7 a^5 z^9-10 a^5 z^7+9 a^5 z^5-15 a^5 z^3+12 a^5 z-3 a^5 z^{-1} +a^4 z^{10}+8 a^4 z^8-22 a^4 z^6+22 a^4 z^4-13 a^4 z^2+3 a^4+3 a^3 z^9-a^3 z^7-9 a^3 z^5+9 a^3 z^3-2 a^3 z-a^3 z^{-1} +4 a^2 z^8-9 a^2 z^6+5 a^2 z^4-a^2 z^2+a^2+3 a z^7-9 a z^5+8 a z^3-2 a z+z^6-3 z^4+2 z^2 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
j \
4           1-1
2          2 2
0         41 -3
-2        72  5
-4       95   -4
-6      96    3
-8     99     0
-10    79      -2
-12   59       4
-14  37        -4
-16  5         5
-1813          -2
-201           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-8 {\mathbb Z} {\mathbb Z}
r=-7 {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.